We introduce and analyze an efficient numerical homogenization method for a class of nonlinear parabolic problems of monotone type in highly oscillatory media. The new scheme avoids costly Newton iterations and is linear at both the macroscopic and the microscopic scales. It can be interpreted as a linearized version of a standard nonlinear homogenization method. We prove the stability of the method and derive optimal a priori error estimates which are fully discrete in time and space. Numerical experiments confirm the error bounds and illustrate the efficiency of the method for various nonlinear problems.